Generalized Method of Eigenoscillations in Diffraction Theory. With 24 figures and 11 tables

The book presents a new method for solving various diffraction and scattering problems in acoustics, electrodynamics, and quantum mechanics. Each version of the method is based on the representation of the diffracted field in the form of a series in the eigenfunctions of an auxiliary homogeneous problem in which the spectral parameter is usually not the frequency. This allows one to treat problems not only in bounded but also in unbounded domains or in the entire space. For example, for the problem of diffraction on a metallic body, the homogeneous problem of the same form can be used with impedance as the spectral parameter.
The transparency coefficient, the dielectric constant, etc. can also be used as the spectral parameter. The method is especially effective for the analysis of resonance systems, in particular, of open resonators and waveguides. The method permits one to represent the exact solution in unbounded domains in the form of a series (since the spectrum is discrete), without an additional integral with respect to the spectral parameter, and use the variational approach though the corresponding problems are usually nonself-adjoint.

M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, N. N. Voitovich Generalized Method of Eigenoscillations in Diffraction Theory The book presents a new method for solving various diffraction and scattering problems in acoustics, electrodynamics, and quantum mechanics. Each version of the method is based on the representation of the diffracted field in the form of a series in the eigenfunctions of an auxiliary homogeneous problem in which the spectral parameter is usually not the frequency. This allows one to treat problems not only in bounded but also in unbounded domains or in the entire space. For example, for the problem of diffraction on a metallic body, the homogeneous problem of the same form can be used with impedance as the spectral parameter. The transparency coefficient, the dielectric constant, etc. can also be used as the spectral parameter. The method is especially effective for the analysis of resonance systems, in particular, of open resonators and waveguides. The method permits one to represent the exact solution in unbounded domains in the form of a series (since the spectrum is discrete), without an additional integral with respect to the spectral parameter, and use the variational approach though the corresponding problems are usually nonselfadjoint. The formal exposition of the method is presented in Chapters 1 and 2. The variational approach is described and analysed in Chapter 3. Chapter 4 contains a number of examples with applications of the method to particular diffraction problems. Chapter 5 contains a rigorous mathematical treatment of the main versions of the method on the basis of modern tools of the theories of nonself-adjoint operators and elliptic pseudodifferential operators. This investigation provides more deep information about the properties of classical integral and integro-differential operators related to the Helmholtz equation and the Maxwell system than in well-known textbooks.